Articles of algebra precalculus

no. of Digit in $x^y\;,$ where $x,y\in \mathbb{N}$

$(1)$:: Calculation of no. of Digits in $2^{100}$ .$(2)$:: Calculation of no. of Digits in $3^{100}$. If it is given that $\log_{10}(2)=0.3010$ and $\log_{10}(3) = 0.4771$ $\bf{My\; Try::}$ I have seen in book and it is given as :: $(1)$ no. of Digit in $\displaystyle 2^{100}$ is equal to $\displaystyle \lfloor \log_{10}(2)^{100}\rfloor +1\;,$ where $\lfloor […]

How to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?

Based on this question i asked recently: A question about geometry of plane curve books, i think it is too advance for a HS student/ typical second or third year undergraduate math majors to read on their own on the books given on the answer to that question Also, i think it is too much […]

Need help solving linear equations with elimination and substiution method

I need help solving system of equations. I never worked with these equations before, so I’m not sure how this works. If anyone could show me the steps so I could go from there, that would be great. Anyways, I need to use substitution method to solve x – 3/2y = 1 2x -7y = […]

Proof by induction that $(a^n-1)$ is divisible by $(a-1)$

I have to proof that $(a^n-1)$ is divisible by $(a-1)$ where $a \in \mathbb {N_{>1}}$ I think that I have the proof but I am not sure if that is the correct format. Induction hypothesis: $(a^n-1)=x(a-1)$ where $x \in \mathbb {N}$ When $n=0$ and $x=0$ each side will be $0$ To be proofed: $(a^{n+1}-1)=m(a-1)$ where […]

Taylor expansion of an stirling identity

I have been searching many ways for a week just to solve this, to no avail. I’m still confused about how the Taylor expansion is produced. It is so advanced compared to the subjects that I took. I am currently taking advance researches or work/journals from other mathematicians but I still cannot do this: $$\frac{(e^{w}-1)^{k}}{k!} […]

If $x$ is positive, then why does $\frac{1}{\sqrt{x+1} + \sqrt{x}} = \sqrt{x+1} – \sqrt{x}$?

Given that $x$ is positive, $\frac{1}{\sqrt{x+1} + \sqrt{x}} = \sqrt{x+1} – \sqrt{x}$ I’ve been trying to convert the left side of the equation to the right side: $$ \frac{1}{\sqrt{x+1} + \sqrt{x}}$$ But then how can I flip this round to be what I have on the right side? I know that $\frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}$, so […]

How to rearrange formulas to calculate orbit from tangent and apoapsis

I need some help rearranging some orbital mechanics formulas. All images have been borrowed from which has a through treatment of orbital equations, but provides no further insight on my question. Given the following: I have $r$, $\phi$ and $R_{a}$ or $R_{p}$ (only one will be relevant in each instance). I need to derive […]

Notation: is it correct to state $3a=a3$?

If $a$ is a real constant, do you regard $3a$ and $a3$ as equal or different?

Probability that team $A$ has more points than team $B$

Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The […]

Quadratic formula – math error

I’m attempting a past paper and I have been asked to compute the derivative for $(x^2-2x+2)$ and from this I calculated $2x-2$. Once I completed this, I was then asked to find and classify the stationary point I usually use quadratic formulas to start this off, but for some reason I’m receiving a maths error. […]